Shashank Gupta: December 2012

The objective of Project 3 is to construct Eigenspace manifolds of images and to use it for object identification and pose estimation. For this purpose, a set of training images of different objects depicted in Figure 1(left) had been captured. A set of test images were also captured in the similar fashion (i.e, they have the same statistics).

Figure 1: Images of objects used in this project for classification(Left). Each objectis acquired by placing them at the center of an imaging sphere and sampling lines of constant co-latitude [right].

Computing and analyzing the Eigenspace:

In this part of the project, the eigenvectors of the images of different objects were calculated and projected in the eigenspace. In order to construct local manifolds, a set of 128 training images of every single objects were considers. Local Eigenspace manifolds for some of the objects are given below.

Local manifold for object Boat

Local manifold for Cabinet


Local manifold for Cup

The manifolds tells us about the distribution of training images on Eigenspace. If we look at the local manifold of cabinet, we observe that most of the part of manifold overlaps, which is due to the fact that the cabinet looks alike from different orientation.

The principle eigenimages for different images are given below.

Principle Eigenimages of Boat


Principle Eigenimages of Cabinet

Principle Eigenimages of Cup

Using the relation of Energy recovery ratio, the subspace dimensions were calculated for the desired level of accuracy. If we plot a plot a graph of energy contained by the eigenimages versus the number of subspace dimensions, we can see that most of the energy can be recovered by considering lower no. of subspace dimensions.

Energy recovery versus subspace dimension curve for Boat

Energy recovery versus subspace dimension curve for Cabinet

A table containing the list of subspace dimensions for different objects is given below.

Calculation of subspace dimension for different energy recovery.

In order to construct global manifold, all the training images of different objects were considered at once. It was basically done by two methods.

Concatenate all 2B images for each of the n objects into a matrix G and
compute the eigenspace of this G.
Concatenate each of the kn-dimensional eigenspaces into a matrix L and
compute the eigenspace of L.

The global manifolds for both the methods are given below.

Global manifold using G matrix

Global manifold using L matrix

The principle Eigenimages for global manifolds are given below.

Global Eigenimages using G matrix

Global Eigenimages using L matrix

The energy recovery plot of global eigenspace are given below.

Energy recovery versus subspace dimension curve for Global manifold(using G matrix)

Energy recovery versus subspace dimension curve for Global manifold(using L matrix)

A plot showing the projection of test images on a local manifold is given below. Since the test images are also captured under the same statistics, it falls close to the points on the manifold. Then we find out the minimum distance of the projected point to the points on the manifold. The one with the least distance is correct match.

A screen shot of the user interaction part of the program is shown below.

Here are the some of the example of projected test image and the match result with pose estimation.

For the Global manifold, L matrix decreases computational load but accuracy is comparatively higher in G matrix approach.

An attempt to map the appearance manifold to some low dimensional manifold with some desirable geometric structure was made. In this case I tried to map the manifold on a two dimensional circle. By doing so, it allows for the classification of the object to be computed using simple calculation compare to the exhaustive search that we performed in the previous case. This drastically decreases the computational cost.
Results are given below.